ENV Envisioning Information Lecture 12 – Scientific Visualization Scalar 2D Data Ken Brodlie

ENV D Interpolation – Regular Gridded Data Suppose we are given data on a regular rectangular grid f given at each grid point; we fill out the empty spaces by interpolating values within each cell

ENV Straightforward extension from 1D: take f-value from nearest data sample No continuity Bounds fixed at data extremes Nearest Neighbour Interpolation

ENV Consider one grid rectangle: –suppose corners are at (0,0), (1,0), (1,1), (0,1)... ie a unit square –values at corners are f 00, f 10, f 11, f 01 f 00 f 10 f 01 f 11 How do we estimate value at a point (x,y) inside the square? Bilinear Interpolation

ENV f 00 f 10 f 01 f 11 (i) interpolate in x-direction between f 00,f 10 ; and f 01,f 11 (ii) interpolate in y-direction We carry out three 1D interpolations: Exercise: Show this is equivalent to calculating f(x,y) = (1-x)(1-y)f 00 +x(1-y)f 10 +(1-x)yf 01 + xyf 11 (x,y) Bilinear Interpolation

ENV Apply within each grid rectangle Fast Continuity of value, not slope (C 0 ) Bounds fixed at data extremes Piecewise Bilinear Interpolation

ENV Contour Drawing Contouring is very common technique for 2D scalar data Isolines join points of equal value –sometimes with shading added How can we quickly and accurately draw these isolines?

ENV As an example, consider this data: Where does the zero level contour go? An Example

ENV The bilinear interpolant is linear along any edge - thus we can predict where the contour will cut the edges (just by simple proportions) cross-section view along top edge Intersections with sides

ENV Simple Approach A simple approach to get the contour inside the grid rectangle is just to join up the intersection points Question: Does this always work? Try an example where one pair of opposite corners are positive, other pair negative

ENV But this does not always work - look at this data: Try it - it is ambiguous! Ambiguity

ENV We need to worry about the behaviour of the interpolant inside the grid square The contour of the bilinear interpolant is NOT a straight line – it is a curve This is curve of: f(x,y) = (1-x)(1-y)f 00 +x(1-y)f 10 +(1-x)yf 01 + xyf 11 = 0 BUT how can we draw it? Joining intersections with straight lines was only an approximation… drawing the contour of bilinear interpolant will resolve ambiguity What is the Problem?

ENV Tracking Contours We can track the contour in small steps through the grid rectangle – starting from intersection with the edges –Take a step, probe at equal distance to either side, then predict next point Next point on contour Current point on contour Probes BUT THIS IS SLOW!! Can we find an alternative which uses straight lines, but resolves the ambiguous case?

ENV Implementing Rectangle-based Contouring For a rectangle, there will be 2 4 = 16 cases There are 4 configurations –All same sign (no contour) –3 same sign (one contour piece) –2 adjacent with same sign (one contour piece) –2 opposite with same sign (two pieces, but ambiguous… function has a saddle) Note: the ambiguity can be resolved by looking at value at saddle point: Saddle value = (f 00 f 11 - f 01 f 10 )/ (f 00 +f 11 -f 01 +f 10 )

ENV Solution by Decomposing Cell Another possibility is to split cell into four triangles Within a triangle, we can fit a linear model –F(x,y) = a + bx +cy How do we split? How do we calculate a,b,c? What is the gain? f1f1 f2f2 f3f3

ENV Cell Decomposition Problem of drawing the curved lines has been circumvented by decomposing cell into four pieces within which the contours are well defined straight lines How might we estimate value at centre?

ENV Contouring in IRIS Explorer Contour is a simple contouring module NAGContour interpolates the data and then tracks the contours NAGContour

ENV Often the data will be given, not on a regular grid, but at scattered locations: f given at each marked point Approach: (i) triangulate (ii) build interpolant in each triangle (iii) draw contours 2D Interpolation – Scattered Data

ENV Triangulation is the process of forming a grid of triangles from the data points How can we construct the triangulation? Triangulation

ENV We solve the DUAL problem: –Suppose a wolf is stationed at each data point. Each wolf is equally powerful and dominates the territory closest to its own base –What are the territories dominated by each wolf? Tesselation

ENV Tesselation – Two or three wolves

ENV Tesselation - Two or Three Wolves

ENV The resulting tesselation is known as the Dirichlet or Voronoi tesselation Given the Dirichlet tesselation for N points P 1, P 2,... P N there is an algorithm for constructing the tesselation when an extra point is added Dirichlet Tesselation

ENV P1P1 P2P2 P3P3 Tesselation for P 1, P 2, P 3 Q Point Q added Dirichlet Tesselation

ENV P1P1 P2P2 P3P3 Q Dirichlet Tesselation

ENV P1P1 P2P2 P3P3 Q Dirichlet Tesselation

ENV –Determine polygon containing Q - here D 3, surrounding P 3 –Construct perpendicular bisector of P 3 Q and find intersection with D 3 - this becomes point of modified tesselation –Determine adjacent polygon - here D 2 –Repeat the above two steps until D 3 is reached again, or there is no intersection –Remove all vertices and edges interior to the new polygon Dirichlet Tesselation

ENV P1P1 P2P2 P3P3 Q Delaunay Triangulation

ENV Triangulation formed by joining points whose territories share a common boundary in the tesselation This has the nice property that it avoids long skinny triangles See the nice applets at: Delaunay.html Note the empty circle property of the Delaunay triangulation Delaunay Triangulation

ENV Contouring from Triangulated Data The final step is to contour from the triangulated data Easy – because contours of linear interpolant are straight lines – see earlier

ENV Implementing Triangle-based Contouring Each vertex can be positive or negative (ignore zero for now) This gives 2 3 = 8 possible cases… … but there are only 2 distinct configurations –No contour (all same sign) –Contour (2 of one sign, 1 of the other) Implementation: –Determine which of 8 cases –Select code for the appropriate configuration f1f1 f2f2 f3f3 f1f1 f2f2 f3f3 All same sign Two same sign In IRIS Explorer, NAGContour will contour scattered data – using triangulation followed by a tracking method based on a nonlinear interpolant – for greater accuracy

ENV A different mapping technique for 2D scalar data is the surface view. Here a surface is created in 3D space, the height representing the scalar value Construction is quite easy - suppose we have a rectangular grid Surface Views

ENV Constructing a Surface View - 1

ENV Surface created as pair of triangles per grid rectangle. Rendering step is then display of triangles. Constructing a Surface View - 2

ENV Examples - with added contours

ENV A further mapping technique for 2D data is the image plot There are three variants: –dot array : draw a dot at each data point, coloured according to the value (very fast, but low quality) Image Plots

ENV Image Plots Grid lines:

ENV Image Plots Areas:

ENV IRIS Explorer – Image Plots and Surface Views Image views are created using: –LatToGeom: takes a lattice (array of values) and produces a colour mosaic –GenerateColorMap: associates colours with values Surface views are created using: –DisplaceLat: takes two lattices, one to create the surface (displacement), the other to colour it (function)… and builds a 3D lattice –LatToGeom: converts this 3D lattice to geometry

ENV Module Assessment Research topic (50) Portfolio to include: –November draft submission –Review comments –Response to comments –Final paper (35) –Presentation (15) Practical exam (50) –One exercise using xmdvtool (25) –One exercise using IRIS Explorer (25) –Task will be to explore a dataset in each case –2 hour supervised exam in laboratory